IOP PUBLISHING BIOINSPIRATION & BIOMIMETICS

Bioinsp. Biomim. 3 (2008) 026001 (11pp) doi:10.1088/1748-3182/3/2/026001

Resonant hopping of a robot controlled byan artificial neural oscillatorEvan H Pelc1, Monica A Daley2,3 and Daniel P Ferris1,2,4

1 Department of Biomedical Engineering, University of Michigan, Ann Arbor, MI 48109, USA2 Division of Kinesiology, University of Michigan, Ann Arbor, MI 48109, USA3 Structure and Motion Laboratory, Royal Veterinary College, Hatfield, Hertfordshire, AL9 7TA, UK4 Department of Physical Medicine and Rehabilitation, University of Michigan, Ann Arbor, MI 48109,USA

E-mail: ferrisdp@umich.edu

Received 14 July 2007Accepted for publication 22 February 2008Published 27 March 2008Online at stacks.iop.org/BB/3/026001

AbstractThe bouncing gaits of terrestrial animals (hopping, running, trotting) can be modeled as ahybrid dynamic system, with spring-mass dynamics during stance and ballistic motion duringthe aerial phase. We used a simple hopping robot controlled by an artificial neural oscillator totest the ability of the neural oscillator to adaptively drive this hybrid dynamic system. Therobot had a single joint, actuated by an artificial pneumatic muscle in series with a tendonspring. We examined how the oscillator-robot system responded to variation in two neuralcontrol parameters: descending neural drive and neuromuscular gain. We also tested theability of the oscillator-robot system to adapt to variations in mechanical properties bychanging the series and parallel spring stiffnesses. Across a 100-fold variation in bothsupraspinal gain and muscle gain, hopping frequency changed by less than 10%. The neuraloscillator consistently drove the system at the resonant half-period for the stance phase, andadapted to a new resonant half-period when the muscle series and parallel stiffnesses werealtered. Passive cycling of elastic energy in the tendon accounted for 70–79% of themechanical work done during each hop cycle. Our results demonstrate that hopping dynamicswere largely determined by the intrinsic properties of the mechanical system, not the specificchoice of neural oscillator parameters. The findings provide the first evidence that an artificialneural oscillator will drive a hybrid dynamic system at partial resonance.

(Some figures in this article are in colour only in the electronic version)

Introduction

In hopping, running and trotting gaits, humans and allterrestrial animals use a similar ‘bouncing’ motion foreconomic locomotion [1–3]. A simple model describesthe fundamental characteristics of these bouncing gaitsremarkably well. The model is a hybrid dynamic system,comprising spring-mass dynamics during stance and ballisticmotion during the aerial phases [4]. The stance phase requiresthe majority of muscle force and work to redirect the motion ofthe body. Consequently, most of the metabolic energy cost oflocomotion is also associated with stance [5–8]. Spring-massdynamics during stance can improve locomotor economy bycycling mechanical energy in elastic tissues and minimizing

active muscle work [1, 9]. In particular, long tendons of distallimb muscles, such as the Achilles tendon in humans, play akey role in storing and returning elastic energy during bouncinggaits [10].

Humans can adjust their overall effective leg stiffnessto hop or run at different stride frequencies [11, 12], butthey have a preferred leg stiffness when allowed to freelychoose their movement dynamics [11, 13]. When humanhopping or running is perturbed by compliant surfaces (i.e.series compliance) [14–16] or elastic leg braces (i.e. parallelcompliance) [17], subjects adjust leg stiffness to maintain thesame overall effective system stiffness. This results in a similarcontact period across conditions. Some have argued that thecontact period choice is related to neural reflex dynamics

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Bioinsp. Biomim. 3 (2008) 026001 E H Pelc et al

[18, 19], but other studies have pointed toward a mechanicalexplanation [20–22]. When a damped spring-mass systemis rebounding off the ground, there is a time of contact withthe surface that requires the least amount of energy input tomaintain steady state because it takes advantage of the resonantfrequency of vibration [22]. Driving an oscillating mechanicalsystem at its natural frequency requires less mechanical energyinput than driving the system at alternative frequencies. Therelative importance of mechanical and neural factors in shapinghopping and running dynamics is unclear.

The neuromechanical control of locomotion is facilitatedby rhythmic neural networks known as central patterngenerators. Central pattern generators are collectionsof neurons that generate the basic pattern for rhythmicmovements. They exhibit common behavioral features suchas the generation of rhythmic motor output in the absenceof rhythmic input and entrainment to sensory feedback [23,24]. These basic features are also demonstrated by artificialneural oscillators, systems of nonlinear equations that exhibitoscillatory behavior. Researchers have used a plethora ofdifferent mathematical equations to model neural oscillatorfunction [25–33]. The behavior produced when coupling theseartificial neural oscillators to a mechanical system is similar formany different specific formulations, suggesting that there aregeneral characteristics of rhythmic neuromechanical systems[34–37].

Several key features of artificial neural oscillators makethem desirable for bio-inspired control of robots. They providerobust and versatile control over a wide range of conditions.Neural oscillators produce stable limit cycles that are robustto transient perturbations yet flexible in adapting to sensoryfeedback [38, 39]. As a result, they can generate locomotionthat automatically changes gait and adjusts to perturbations tomove effectively through a variable environment [25, 26, 28,40–43]. Importantly, neural oscillators entrain to the dynamicsof the mechanical system, driving the system at its resonantfrequency [33, 34, 36, 37]. Theoretically, this could minimizethe energetic cost of locomotion for legged robots. It alsoreduces the need for trajectory planning and precise knowledgeof mechanical system properties (e.g., mass, stiffness).

Although several studies have reported how artificialneural oscillators entrain to continuous dynamic systems (suchas pendulums), their performance in driving discontinuoushybrid dynamic systems (i.e. spring-mass systems withseparate stance and aerial phases) is not established. Hybriddynamic systems do not have a single resonant frequencybecause their dynamics shift between governing sets ofequations. The dynamics of hybrid systems can bequite complex, prohibiting complete analytical solutionsin mathematical models unless many simplifications andconstraints are applied [44]. Ijspeert and colleaguesdemonstrated that an artificial neural oscillator (i.e. Hopfoscillator) can drive a simulated spring-mass hopper withhybrid dynamics [45]. They also found that the adaptiveartificial neural oscillator adopts a hopping frequency thatmaximizes a measure of hopper economy [45]. However,they did not perform a thorough mechanical analysis of theirsimulation so it is unclear how the neural oscillator interactedwith the mechanical dynamics.

We built a simple hopping robot with hybrid dynamics sothat we could test the ability of an artificial neural oscillator toadaptively entrain to a hybrid system. Our neuromechanicalsystem consisted of a single reciprocal inhibition oscillator(Matsuoka oscillator) coupled to a hopping robot actuated ata single ankle joint. We used two approaches to test theadaptive capabilities of the artificial neural oscillator. First,we examined whether the entrainment of the oscillator-robotsystem is robust across changes in two control parameters:descending neural drive and muscle gain. Second, we testedthe ability of the neuromechanical system to adapt to variationin mechanical properties by altering the resonant frequency ofthe stance phase via changes in the in-series and in-parallelspring stiffnesses. We hypothesized that the artificial neuraloscillator would drive the mechanical system in resonance forthe ground contact phase. The aerial phase has no resonancedue to the dynamics of projectile motion.

We chose to test the system response to descending neuraldrive and muscle gain because these are two control parameterswith well-understood physiological analogs. In vertebrateanimals, locomotion is initiated by descending neural drivefrom the mesencephalic locomotor region (MLR) of themidbrain [46, 47]. Increased strength of the descending signalfrom the MLR results in increased frequency and amplitude ofrhythmic motor output with abrupt transitions between gaits.In this manner, the brain can govern speed, amplitude andmode of movement. In our simple neuromechanical system,altering the descending signal strength was a way to test ifthe robot maintained similar hopping dynamics with changesin general excitation. The second control signal, musclegain, represents the excitation–contraction coupling betweenthe nervous system and muscle. This relationship can varyunder physiological conditions such as muscle fatigue [48,49]. By altering the muscle gain, we examined the abilityof the oscillator to adapt to changes in the neuromechanicaltransfer dynamics. With this parameter, we could test whetherthe system maintains similar hopping dynamics with changesin the neuromuscular gain.

Methods

Hopper design

We designed the robot to represent general aspects of lowerlimb mechanics during human hopping. It was composed ofwooden foot and shank segments joined by a metal hinge jointwith one degree of freedom to simulate a simplified ankle.While a hinge joint does not match the true complexity ofthe ankle joint, it replicates the primary motion of this jointin the sagittal plane during hopping. Neoprene padding onthe foot section provided better grip upon landing and lightdamping to diminish high frequency vibration. This createda more biologically realistic model of the human foot, whichemploys considerable tissue padding to cushion foot impact[50]. The robot is stabilized in the medio-lateral plane throughits attachment to a wooden supporting beam to limit motion tovertical hopping (figure 1). The top of the shank was secured tothe beam by a hinge joint, allowing free motion in the sagittal

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Artificial Muscle

Foot

Ankle Joint

Metal Tendon

Elastic Dorsiflexor

Force Sensor

Shank

Beam

Figure 1. The hopping robot. The artificial muscle was connectedto a pneumatic system which regulated air flow into and out of themuscle. The ‘tendon’ was a metal extension spring. The ankle was ametal hinge joint with a single degree of freedom allowing motionin the sagittal plane.

plane. The beam was attached to a static base through anotherhinge joint.

The ankle joint was actuated for plantar flexion by anartificial muscle–tendon unit, consisting of a pneumatic muscleand metal extension spring connected in series. The musclewas attached to the proximal shank by an L-bracket and thespring (Century Spring Corp., Los Angeles, CA) insertedvia an eyebolt onto the superior, posterior aspect of the foot(figure 1). A single-axis compression load cell (Omegadyne,Sunbury, OH), placed between the L-bracket attachment andthe superior end of the muscle, allowed measurement ofmuscle–tendon force over the course of each hop cycle.While the artificial muscle–tendon unit provided ankle plantarflexion, antagonistic dorsiflexion was mediated by a passivespring (elastic band) attached to the anterior side of the shankand foot. We did not include active dorsiflexion for the robotbecause we wanted the simplest system that could test ourhypothesis. Pilot data collections with the robot including anartificial pneumatic muscle controlled by the flexor half-centerof the artificial neural oscillator demonstrated qualitativelysimilar behavior as when an elastic rubber band was used forpassive dorsiflexion.

The artificial pneumatic muscle was constructed in thesame manner as that used for previous studies on pneumaticallypowered ankle–foot orthoses [51–53]. The body of themuscle consisted of latex tubing for the inner air bladder andbraided polyester sheathing for the outer shell. Four parallelproportional pressure regulators (MAC Valves, Inc., Wixom,MI) supplied compressed air (0–6.2 bar) to each artificialmuscle via nylon tubing. An analog-controlled solenoid valve(MAC Valves, Inc., Wixom, MI) was placed in parallel withthe air supply tubing to facilitate exhaust. Air was suppliedto the artificial muscle through a regulated air supply throughstandard pneumatic fittings. We controlled the air pressure to

Reciprocal Inhibition

1

2

Recurrent Inhibition

Recurrent Inhibition

Output:

Artificial Plantarflexor

Output:

Dorsiflexor

(Inactive)Sensory

Feedback Input

Tonic Descending

Input

1y

2y

k1vβ

2vβ

2yη1yη

][ jj ghjg

+][ jj gh

Figure 2. Artificial neural oscillator schematic. Ankle angle servedas the sensory feedback signal. Oscillator output included bothflexor and extensor muscle activation signals. For our testing, onlythe plantar flexor oscillator half-center was used. Dorsiflexion onthe robot was achieved by a passive rubber spring.

the muscle through a real-time computer interface (dSPACEInc., Northville, MI; 1000 Hz) that received its control signalfrom the neural oscillator output. The artificial muscle had asimilar force bandwidth (2.4 Hz) as human skeletal muscle(2.2 Hz) [52, 54]. The effective mass of the robot wasapproximately 3.6 kg.

Artificial neural oscillator

The neural circuitry of a central pattern generator (CPG)was modeled using the dynamic equations for a simplifiedreciprocal inhibition oscillator developed by Matsuoka(figure 2) [31, 55]. The neural oscillator model was governedby the following set of differential equations:

τ1x1 = −x1 − βv1 − η[x2]+ + k − hj [gj ]+

τ2v1 = −v1 + [x1]+

τ1x2 = −x2 − βv2 − η[x1]+ + k − hj [gj ]−

τ2v2 = −v2 + [x2]+

yi = [xi]+

Ti = Giyi,

where τ 1 and τ 2 are time constants of the inner state andadaptation effect, respectively, which adjust the shape andintrinsic frequency of the oscillator, η is a constant that affectsthe reciprocal inhibition between the two neurons, β is aconstant involved in the recurrent inhibition of each neuron, kis the tonic descending signal strength and g is the feedbackinput (i.e. joint angle) with a corresponding gain h. Thevalue xi is a discontinuous term, where [xi]+ is the positivepart of xi and used as the output signal from the oscillator,yi. T is the final control signal (i.e. voltage) sent to thepressure regulator and is the product of yi and muscle gain,G. This control signal can be thought of as analogous to anelectromyography linear envelope [56]. The base value foreach of the model parameters was chosen using guidelinesdeveloped by Williamson through a rigorous analysis of the

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Table 1. Base parameter values used for the neural oscillator(unitless).

Parameter Value

τ 1 0.02τ 2 0.04β 2.5k 3.0h 5.0G 10.0

controller properties [57, 58] (table 1). The values of τ 1

and τ 2 were set to provide an intrinsic oscillator frequencyapproximately one-half the resonant mechanical frequency ofthe hopper in the standard spring configuration. The preciseintrinsic frequency of the oscillator is unlikely to influencethe results, however, because this oscillator has been foundto robustly entrain to the mechanical system over a widefrequency range [58]. The tonic descending signal representsdescending neural drive from the midbrain [46, 47]. Basedon the assumption that k would influence system dynamicsin a manner consistent with biological data, we expectedincreased descending drive to produce increased frequencyand/or amplitude of motor output. Muscle gain (G) representsthe neuromechanical transformation between neural signal andcontraction dynamics.

We used the Matsuoka artificial neural oscillator becauseof its simplicity, demonstrated robustness and adaptability, andwide use in neuromechanical modeling and adaptive robotics[25, 31, 37–43, 55, 57, 59]. However, as discussed in theintroduction, previous experiments have found similar systembehavior using a number of different oscillators.

Experimental procedure

Two separate experimental manipulations were performed toevaluate the ability of the neural oscillator to provide robust andadaptive control of the hopping robot. First, we manipulatedoscillator control parameters to examine the sensitivity ofthe neuromechancial system. Second, we manipulated theproperties of the mechanical system to examine the ability ofthe neural oscillator to drive the system over a range of stance-phase resonant frequencies. We varied the stiffness of themuscle–tendon unit by altering the arrangement of in-seriesand in-parallel springs, described in detail below.

In the first experimental manipulation, we varied the tonicdescending signal strength (k) and muscle gain (G) parametersand analyzed the resulting system performance. The standardspring setup was used as depicted in figure 1 (tendon springof 2.7 kN m−1). The k and G values were individually variedover a range of 0.1 to 100. The value of each parameter wasincreased exponentially, resulting in at least ten trials for eachvaried parameter. For each trial, all model parameters werefixed except the current variable under investigation (either k orG) (table 1). With increasing G, the hopping dynamics rapidlyreached a consistent pattern that was robust to further changesin G. Because tonic drive (k) exhibited a less sharp saturation,more values of k were tested to resolve the point at which thehopping dynamics reached a consistent pattern. Kinetic and

kinematic data were collected for five continuous seconds afterallowing the robot to first reach steady state hopping.

In the second experiment we examined systemperformance under conditions of varying muscle–tendonstiffness, manipulated by changing the arrangement of springsin parallel and in series. First, we tested four different springs(1.7, 2.7, 4.1, 6.7 kN m−1) in the standard arrangement,changing the in-series stiffness (figure 1). The springs wereof equal length and differed minimally in mass (0.147, 0.16,0.19, 0.19 kg). In a second spring arrangement, we used the2.7 kN m−1 spring in series with the artificial muscle andadded one of the four springs of differing stiffness in parallel.The second spring was attached (via rigid cables) in parallelto the muscle–tendon unit, so that the two were positionedbisymmetrically about the center point on the posterior side ofthe shank section. This arrangement of the parallel spring wasintended to mimic previous experiments of humans hoppingin place with an elastic ankle–foot orthosis [17].

Data collection

An eight-camera motion analysis system (Motion Analysis,Santa Rosa, CA) was used to record segment and jointkinematics at 120 Hz. We placed reflective markers on thetoe and heel of the foot, at the proximal and distal ends ofthe artificial muscle, at the center of rotation of the ankle jointand in a triad of markers on the anterior face of the shank.We measured the ankle angle in the sagittal plane using anelectrogoniometer (Biometric Ltd, Ladysmith, VA). This angleserved as the neural oscillator feedback signal, which was thesole input to the computer controller. We positioned the hopperdevice over a force plate and collected ground reaction force(GRF) data at 1200 Hz (Advanced Mechanical Technology,Watertown, MA).

Data analysis

The data were filtered and processed using commercialsoftware programs (Visual3D, C-Motion, Rockville, MD;MATLAB, Mathworks). Data were filtered using a fourth-order zero-lag Butterworth low-pass filter with a 15 Hz cutofffrequency. Individual hop cycles were defined as the timefrom ground contact to subsequent ground contact, determinedfrom the GRF data. The data for all complete hop cycles ineach 5 s trial were aligned and averaged at each time point togenerate a single average hop cycle. Incomplete hop cycleswere excluded from analysis. The number of complete hopcycles in the 5 s recording period varied between 9 and 14depending on hop frequency.

Calculations

For each hopping cycle, we measured and calculated anumber of traditional biomechanical variables. We determinedhopping frequency, stance duration and aerial phase durationfrom the ground reaction force data. We calculated thedisplacement of the tendon spring by dividing the muscle–tendon force (measured with the load cell) by the springstiffness. We used kinematic markers on the proximal and

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Bioinsp. Biomim. 3 (2008) 026001 E H Pelc et al

period period

(A) (B)

AnkleAngle

Time

Figure 3. Measurements of the half-period of vibration for passive free vibration without an aerial phase and for active hopping.(A) Example of amplitude of ankle angle versus time in free vibration testing. The time duration of a given cycle half-period is the value ofinterest. (B) Example of vertical ground reaction force (VGRF) versus time during hopping. Points are identified on each force curve wherethe data equal bodyweight of the robot. The half-period of vibration during the stance phase is the period of time that the ground reactionforce is above the bodyweight threshold.

distal ends of the muscle to record muscle displacement. Wecalculated the mechanical power of the muscle and tendon bymultiplying their respective velocities by the muscle–tendonforce. By integrating the power curve with respect to time,we calculated mechanical work. To evaluate the efficiencyof the system, we calculated the fraction of total mechanicalwork done passively through energy cycling in the spring andactively by the artificial muscle.

To test whether the artificial neural oscillator drovethe hopper at the stance phase resonant frequency, wecompared the half-period of the stance phase (time durationfor bodyweight to bodyweight in the ground reactionforce) with the half-period measured during free vibration(figure 3). The mathematics for determining the half-period ofthe stance phase is explained more extensively by Cavagnaet al [60]. The mathematics for determining the half-period of free vibration is explained more extensively byTongue [61].

Results

Changes in the tonic descending drive (k) and muscle gain(G) had similar qualitative effects on hopping mechanics. Atvalues lower than 0.1 for either parameter, the robot did nothop with an aerial phase. Between values of 0.1 and 1, hoppingmechanics were very sensitive to both parameters. The controlsignal generated by the artificial neural oscillator at theseparameter values was of low amplitude and peaked later inthe hop cycle (figures 4 and 5). At slightly higher levels oftonic descending drive and muscle gain, the control signaloccurred earlier in the hop cycle and saturated in amplitude(figures 4 and 5). As a result of the earlier control signalin the hop cycle, the peak muscle–tendon force, peak groundreaction force and duration of the aerial phase were greaterat higher values of tonic descending drive and muscle gain(figures 6 and 7). A longer aerial phase indicates that the robotjumped higher, because the robot undergoes simple ballisticmotion in this phase.

Hopping mechanics showed little variation with changesin descending drive or muscle gain over the parameter rangeof 1–100. For this 100-fold change in parameter values, theartificial muscle developed force while shortening to producepositive power and the tendon underwent stretch and recoil

(figures 4 and 5). Stance duration, hopping frequency, peakforce and displacement of the ankle, muscle and tendonmaintained relatively constant values (figures 6 and 7).Hopping mechanics remain especially constant with alteredmuscle gain (G). Hopping frequency changed by 5% (1.88–1.98 Hz) and ground contact time changed by 4% (0.258–0.269 s) over this 100-fold range in G (figure 7). Increments intonic drive (k) over this range leads to small increments in peakmuscle–tendon force, and corresponding small incrementsin aerial time and decrements in hop frequency (figure 6).However, given the 100-fold range in k, these changes aremodest. Hopping frequency changed by less than 10% (1.85–2.01 Hz) and ground contact time changed by less than 13%(0.234–0.269 s). Over the 100-fold range in both parameters,elastic energy storage in the tendon spring accounted for 70–79% of the total mechanical work done by the muscle–tendonsystem (muscle contributed 21–30%) (figures 6 and 7).

The neural oscillator consistently drove the robot at theresonant half-period of the stance phase. We manipulatedthe resonant half-period by changing the stiffness of the in-series spring (figure 8(A)) and by adding a parallel springof varying stiffnesses (figure 8(B)). This resulted in resonanthalf-periods ranging between 0.13 and 0.19 s for the in-seriesarrangement and 0.12 and 0.16 s for parallel arrangement.The experimentally measured half-period based on groundforce data (figure 3(B)) consistently matched the resonant half-period of the robot. The slope of the least squares linearregression fit between these variables was 1.05 (R2 = 0.93) forthe single-spring setup and 1.02 (R2 = 0.95) for the parallel-spring setup. This indicates that the robot was consistentlydriven at stance phase resonance.

Discussion

Hopping dynamics were primarily dependent on the intrinsicmechanical properties of the robot. Muscle gain andsupraspinal gain were varied over 100-fold, but the hoppingfrequency and ground contact time varied by less than 10 and13%, respectively. The artificial neural oscillator entrainedto the natural frequency of the mechanics to achieve stancephase resonance. When the intrinsic mechanical properties ofthe robot were altered by changing the spring arrangement, the

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Bioinsp. Biomim. 3 (2008) 026001 E H Pelc et al

-2

6

14

0

0

40

80

120

Force(N)

0

100

200

300

Control Signal Ground Reaction Force Load Cell Force

Volts(V)

0 25 50 75 100Time (% Hop Cycle)

0 25 50 75 100Time (% Hop Cycle)

0 25 50 75 100Time (% Hop Cycle)

-200

-100

0

200

300

Muscle and Spring Power

Power (W)

0 25 50 75 100Time (% Hop Cycle)

-0.04

0

0.04

0.08

0.12

0.16 Muscle and Spring Displacement

Length (m)

0 25 50 75 100Time (% Hop Cycle)

(A) (B)

(D) (E )

(C)

k = 0. 1k = 0. 5k = 3k = 10k = 20

100

Force(N)

Figure 4. Effect of descending tonic drive (k) on hop cycle dynamics. (A) Muscle control signal, (B) ground reaction force, (C) load cellforce, (D) muscle and spring displacement and (E) muscle and spring power. In (D) and (E), solid lines are for the spring and dashed linesare for the muscle. As tonic drive increased, the robot progressed toward consistent hop dynamics that were insensitive to further increasesin k. Waveforms are the mean of hops in a 5 s trial. Initial ground contact is at 0% of the hop cycle.

-2

6

14

0

-10

0

25

50

75

100

Force(N)

0

100

200

300

Control Signal Ground Reaction Force Load Cell Force

Volts (V)

0 25 50 75 100Time (% Hop Cycle)

0 25 50 75 100Time (% Hop Cycle)

0 25 50 75 100Time (% Hop Cycle)

-200

-100

0

100

200

Muscle and Spring Power

Power(W)

0 25 50 75 100Time (% Hop Cycle)

-0.04

0

0.04

0.08

0.12

0.16 Muscle and Spring Displacement

Length(m)

0 25 50 75 100Time (% Hop Cycle)

G = 0. 1G = 0. 5G = 3G = 10G = 20Force

(N)

(A) (B)

(D) (E )

(C )

Figure 5. Influence of muscle gain (G) on hop cycle dynamics. (A) Control signal, (B) ground reaction force, (C) load cell force,(D) muscle and spring displacement and (E) muscle and spring power. In (D) and (E), solid lines are for the spring and dashed lines are forthe muscle. The robot progressed toward consistent hop dynamics that were insensitive to further increases in G. Waveforms are the mean ofhops in a 5 s trial. Initial ground contact is at 0% of the hop cycle.

neural oscillator consistently drove the system at the resonanthalf-period of the stance phase (figure 8).

In contrast to our expectations, changes in descendingneural drive had little effect on robot hopping dynamics.

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0

1

2

3

0.1 1 10 100

0

25

50

75

100

0.1 1 10 100

Descending Signal Strength

Spring

Muscle

-0.04

0

0.04

0.08

0.12

0.16

0.1 1 10 100

Spring

Muscle

0

0.1

0.2

0.3

0.4

0.1 1 10 100

Ground Contact

Aerial Phase

0

40

80

120

0.1 1 10 100

0

100

200

300

0.1 1 10 100

0

25

50

75

0.1 1 10 100

Hopping Frequency

(Hz)

Mechanical Work (%)

PeakDisplacement

(m)

Phase Time

(s)

Peak Ground

Reaction Force (N)

PeakLoad Cell Force (N)

Ankle Displacement

(º)

(A)

(B)

(C)

(D)

(E)

(F)

(G)

Figure 6. The effect of descending signal strength on hoppingmechanics. (A) Ground contact and aerial time, (B) hoppingfrequency, (C) ground reaction force, (D) load cell force, (E) ankleangular displacement, (F) spring and muscle displacement and (G)spring and muscle mechanical work. Values for descending signalstrength are displayed on a log10 axis. Values are the means ± s.e.m.

Hopping Frequency

(Hz)

Mechanical Work (%)

PeakDisplacement

(m)

Phase Time

(s)

Peak Ground

Reaction Force (N)

PeakLoad Cell Force (N)

Ankle Displacement

(º)

0

0.1

0.2

0.3

0.4

0.1 1 10 100

Ground Contact

Aerial Phase

0

1

2

3

4

0.1 1 10 100

0

25

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0.1 1 10 100

0

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0.1 1 10 100

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-0.05

0

0.05

0.1

0.15

0.1 1 10 100

Spring

Muscle

0

25

50

75

100

0.1 1 10 100

Muscle Gain

Spring

Muscle

(A)

(B)

(C )

(D)

(E)

(F)

(G)

Figure 7. The effect of muscle gain on hopping mechanics.(A) Ground contact and aerial time, (B) hopping frequency, (C)ground reaction force, (D) load cell force, (E) ankle angulardisplacement, (F) spring and muscle displacement and (G) springand muscle mechanical work. Values for muscle gain are displayedon a log10 axis. Values are the means ± s.e.m.

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Free Vibration Half-Period (s)

StanceHalf-Period (s)

StanceHalf-Period (s)

0.1

0.15

0.2

0.25

0.1 0.15 0.2 0.25

y = 1.05x

y = 1.02x

0.1

0.15

0.2

0.25

0.1 0.15 0.2 0.25

(A)

(B)

Figure 8. Comparison of half-period of vibration during stancephase with the resonant half-period of free vibration. A line withslope of 1 (solid thick line) is the expected relationship if the robotwas driven at stance resonance. The dashed line indicates the leastsquares linear regression fit of the experimental data. (A) Dataacquired using the standard spring setup as shown in figure 1 andchanging the stiffness of the spring tendon. (B) Data acquired usinga parallel spring in addition to the standard setup of the robot,changing the stiffness of the parallel spring. Values are the means ±s.e.m.

Studies on vertebrate locomotion have revealed thatincreases in descending neural drive produce stronger musclecontractions during stance, higher frequency movements andcan lead to abrupt transitions in gait [46, 47, 80]. This hybriddynamic robot does not have a single resonant frequencybecause the dynamics of the stance and aerial phases arediscontinuous. Aerial phase duration depends only on thevertical take-off velocity at the end of stance. We expectedthat increases in neural drive would increase take-off velocity,resulting in a longer aerial phase and lower hopping frequency.Hopping mechanics were sensitive over a small range ofparameter values, but a 100-fold change in the parameter hadlittle effect above the value of 1. The relative consistencyof movement dynamics likely resulted from the constrainedmechanical structure. Our robot represented a simple hoppingsystem consisting of one single degree of freedom joint andone actuator. Although gait changes were not possible, it waspossible to maintain stance phase resonance while alteringaerial phase duration and hopping frequency. It is not clearwhy the system maintained such consistent dynamics acrosschanges in descending neural drive.

The robot greatly relied on elastic energy storage andreturn during hopping. At muscle gain and supraspinal gainparameters greater than 1, the ankle extensor muscle did notchange length for the first half of stance and then contractedto produce positive work during the second half of stance(figures 4 and 5). In contrast, the tendon lengthened duringthe first half of stance and shortened during the second halfof stance (figures 4 and 5). These behaviors are similar tothose of the human triceps surae and Achilles’ tendon duringhopping and running [62–64].

The robot demonstrated an important difference betweenits behavior and that of humans. When humans hop with addedparallel spring stiffness using an ankle–foot orthosis, theymaintain the same ground contact time as they used without thedevice [17]. In contrast, our robot exhibited variable groundcontact times with a similar manipulation. This adjustmentallowed the robot to maintain stance phase resonance. Severalfactors could explain this difference between human androbot, all relating to the greater complexity of the humanneuromechanical system.

Humans might maintain stance phase resonance withadded parallel elasticity by adjusting their movement patternssubtly rather than altering their ground contact time. The morecomplex, nonlinear musculoskeletal dynamics of humans andother animals can be altered by limb posture, muscle co-activation and choices in synergist muscle activation [65–70].The choice of a constant ground contact time both with andwithout an elastic orthosis may have benefits related to muscleactivation dynamics, muscle efficiency or metabolic cost[5, 71–73].

Another potential explanation for the difference betweenhuman and robot is the simplistic model of the robot nervoussystem. Humans, like all vertebrates, have a hierarchicalneural control architecture that integrates brain function, spinalnetworks and reflex feedback [74, 75]. In particular, reflexiveexcitation from stretch and load-dependent proprioceptorscontribute substantially to extensor muscle activation duringstance [18, 19, 77, 78]. There may be an optimal groundcontact time based on the timing of reflex pathways andthe inherent system mechanics (e.g. gravity, body mass)for stability [79]. Our results do not provide a definitiveexplanation for the differences between robot and human, butthey do present possibilities for designing future experiments.

An alternative modeling approach for the analysis ofbouncing gait dynamics would be to rely entirely oncomputer simulations. A key advantage to building an actualphysical device is that it benefits from accurate ‘physicsfor free’. All computer models and simulations rely onsimplifying assumptions about mechanical properties anddynamics. Physical models allow testing and validation ofthese mathematical models, and often reveal aspects of thesystem that otherwise would have been missed [81]. For thisreason, numerous groups have used robots as research toolsfor understanding the basic principles of human and animallocomotor mechanics [82–86]. In addition, the ability to bothsee a system in action and quantify its mechanical performanceoften provides more convincing evidence for the model thansimulation alone. Robotic models serve as an important tool

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for engaging students in research and increasing the impact ofbiomechanical studies, because robots have widespread appealand strong presence in popular culture.

Our biologically inspired robot also has importantimplications for the design and control of robots. Previouswork has demonstrated the robust and adaptive performanceof neural oscillators for controlling simple resonant systems[33, 36–38]. However, few studies have investigated thedynamics of neural oscillators coupled to hybrid dynamicssystems, which do not have a single resonant frequency. Arecent study has demonstrated that a simple adaptive neuraloscillator model can drive a hopping quadrupedal robot atits resonant frequency [45]. However, no previous study hasincluded a detailed analysis of locomotor mechanics whiledriven by a neural oscillator. Our results demonstrate thatthe Matsuoka oscillator drives the mechanical system at theground contact resonance during hopping (figure 8). Analysisof locomotor mechanics reveals that this allows approximately75% of the mechanical work to be done by passive energycycling in the tendon spring (figures 6 and 7). This resultsuggests that neural oscillator control, when appropriatelycoupled to the mechanical system, could improve efficiencyin a wide array of robotic applications. Furthermore, theneural oscillator was able to adapt to the change in springstiffness without the need for hand tuning of control parameters(figure 8). This is consistent with previous studies thatsuggest that neural oscillators produce stabile motor outputthat eliminates the need for trajectory planning and preciseknowledge of mechanical system properties [25, 33, 38, 39,41, 45]. Thus, neural oscillators can provide robust, adaptableand economic control of hybrid dynamic systems.

One potential limitation to artificial neural oscillatorcontrol is that some behaviors require motion at non-resonantfrequencies. However, networks of coupled neural oscillatorscan allow a wider range of behaviors and locomotor modes,with gait transitions automatically controlled by systemdynamics [26, 28]. Additionally, a hierarchical controlstructure could allow the neural oscillator rhythm to beadjusted by higher level control, as observed in animals [74,75]. Such control architectures, although more complex,would still reap the benefits of stable oscillator dynamics:automatic entrainment and integration of sensory feedback,and economic locomotion without trajectory planning.

Conclusion

Biomimetic robots help reveal basic principles ofneuromechanical integration and provide insight intohuman and animal movement. The very simpleneuromechanical hopping system studied here demonstratesthe ability of artificial neural oscillators to provide robust,adaptive and economic control of hybrid dynamic systems.More anthropomorphic robot designs with similar controlarchitecture could be important tools to investigate theneuromechanical integration for dynamically stable running.Yet, addition of multiple joints, actuators and neural oscillatorswill immensely increase the system complexity. Thegreatest insight is gained from neuromechanical models that

incorporate the minimal complexity necessary for the desiredsystem behavior. Consequently, an important goal of futurework will be to identify the minimal neuromechanical systemnecessary to obtain stable forward bouncing locomotion.

Acknowledgments

The authors would like to thank Catherine Kinnaird and TomSerbowicz for their assistance with the project. This work wassupported by an NSF CAREER award to DPF (BES-0347479),an NIH grant to DPF (R01 NS45486), an NSF BioinformaticsPostdoctoral Fellowship to MAD (DBI-0630664) and theUniversity of Michigan UROP.

References

[1] Cavagna G A, Heglund N C and Taylor C R 1977 Mechanicalwork in terrestrial locomotion: two basic mechanisms forminimizing energy expenditure Am. J. Physiol. 233R243–61

[2] Full R J and Farley C T 2000 Musculoskeletal dynamics inrhythmic systems: a comparative approach to leggedlocomotion Biomechanics and Neural Control of Postureand Movement ed J M Winters and P E Crago (New York:Springer-Verlag)

[3] Biewener A A 2006 Patterns of mechanical energy change intetrapod gait: pendula, springs and work J. Exp. Zool.305A 899–911

[4] McMahon T A and Cheng G C 1990 The mechanics ofrunning: how does stiffness couple with speed? J. Biomech.23 65–78

[5] Kram R and Taylor C R 1990 Energetics of running: a newperspective Nature 346 265–7

[6] Kram R 2000 Muscular force or work: what determines themetabolic energy cost of running? Exerc. Sport Sci. Rev. 28138–43

[7] Taylor C R, Heglund N C, McMahon T A and Looney T R1980 Energetic cost of generating muscular force duringrunning: a comparison of small and large animals J. Exp.Biol. 86 9–18

[8] Farley C T and McMahon T A 1992 Energetics of walking andrunning—insights from simulated reduced-gravityexperiments J. Appl. Physiol. 73 2709–12

[9] Alexander R M and Bennet-Clark H C 1977 Storage of elasticstrain-energy in muscle and other tissues Nature 265 114–7

[10] Alexander R M 1988 Elastic Mechanisms in AnimalMovement (Cambridge: Cambridge University Press)

[11] Farley C T, Blickhan R, Saito J and Taylor R C 1991 Hoppingfrequency in humans: a test of how springs set stridefrequency in bouncing gaits J. Appl. Physiol. 71 2127–32

[12] Farley C T and Gonzalez O 1996 Leg stiffness and stridefrequency in human running J. Biomech. 29 181–6

[13] He J P, Kram R and McMahon T A 1991 Mechanics of runningunder simulated low gravity J. Appl. Physiol. 71 863–70

[14] Ferris D P and Farley C T 1997 Interaction of leg stiffness andsurface stiffness during human hopping J. Appl. Physiol. 8215–22

[15] Ferris D P, Louie M and Farley C T 1998 Running in the realworld: adjusting leg stiffness for different surfaces Proc. R.Soc. Lond. 265 989–94

[16] Ferris D P, Liang K and Farley C T 1999 Runners adjust legstiffness for their first step on a new running surfaceJ. Biomech. 32 787–94

[17] Ferris D P, Bohra Z A, Lukos J R and Kinnaird C R 2006Neuromechanical adaptation to hopping with an elasticankle-foot orthosis J. Appl. Physiol. 100 163–70

9

Bioinsp. Biomim. 3 (2008) 026001 E H Pelc et al

[18] Melville-Jones G and Watt D G 1971 Observations on thecontrol of stepping and hopping movements in manJ. Physiol. 219 709–27

[19] Dietz V, Schmidtbleicher D and Noth J 1979 Neuronalmechanisms of human locomotion J. Neurophysiol. 421212–22

[20] Cavagna G A 1970 Elastic bounce of the body J. Appl.Physiol. 29 279–82

[21] Takeshita D, Shibayama A, Muraoka T, Muramatsu T,Nagano A, Fukunaga T and Fukashiro S 2006 Resonance inthe human medial gastrocnemius muscle during cyclicankle bending exercise J. Appl. Physiol. 101 111–8

[22] Bach T M, Chapman A E and Calvert T W 1983 Mechanicalresonance of the human body during voluntary oscillationsabout the ankle joint J. Biomech. 16 85–90

[23] Pearson K G 1993 Common principles of motor control invertebrates and invertebrates Annu. Rev. Neurosci.16 265–97

[24] Grillner S and Wallen P 1985 Central pattern generators forlocomotion, with special reference to vertebrates Annu. Rev.Neurosci. 8 233–61

[25] Taga G, Yamaguchi Y and Shimizu H 1991 Self-organizedcontrol of bipedal locomotion by neural oscillators inunpredictable environment Biol. Cybern. 65 147–59

[26] Collins J J and Richmond S A 1994 Hard-wired central patterngenerators for quadrupedal locomotion Biol. Cybern.71 375–85

[27] Buchli J, Righetti L and Ijspeert A J 2006 Engineeringentrainment and adaptation in limit cycle systems—frombiological inspiration to applications in robotics Biol.Cybern. 95 645–64

[28] Ijspeert A J, Crespi A, Ryczko D and Cabelguen J M 2007From swimming to walking with a salamander robot drivenby a spinal cord model Science 315 1416–20

[29] Kopell N and Ermentrout G B 1988 Coupled oscillators andthe design of central pattern generators Math. Biosci.90 87–109

[30] Ghigliazza R M and Holmes P 2004 A minimal model of acentral pattern generator and motoneurons for insectlocomotion Siam J. Appl. Dyn. Syst. 3 671–700

[31] Matsuoka K 1985 Sustained oscillations generated bymutually inhibiting neurons with adaptation Biol. Cybern.52 367–76

[32] Skinner F K, Kopell N and Marder E 1994 Mechanisms foroscillation and frequency control in reciprocally inhibitorymodel neural networks J. Comput. Neurosci.1 69–87

[33] Hatsopoulos N G 1996 Coupling the neural and physicaldynamics in rhythmic movements Neural Comput. 8567–81

[34] Iwasaki T and Zheng M 2006 Sensory feedback mechanismunderlying entrainment of central pattern generator tomechanical resonance Biol. Cybern. 94 245–61

[35] Sekerli M and Butera R J 2005 Oscillations in a simpleneuromechanical system: underlying mechanismsJ. Comput. Neurosci. 19 181–97

[36] Simoni M F and DeWeerth S P 2007 Sensory feedback in ahalf-center oscillator model IEEE Trans. Biomed. Eng.54 193–204

[37] Verdaasdonk B W, Koopman H and Van Der Helm F C T 2006Energy efficient and robust rhythmic limb movement bycentral pattern generators Neural Netw. 19 388–400

[38] Verdaasdonk B W, Koopman H and Van der Helm F C T 2007Resonance tuning in a neuro-musculo-skeletal model of theforearm Biol. Cybern. 96 165–80

[39] Williamson M M 1998 Neural control of rhythmic armmovements Neural Netw. 11 1379–94

[40] Taga G 1994 Emergence of bipedal locomotion throughentrainment among the neuro-musculo-skeletal system andthe environment Physica D 75 190–208

[41] Taga G 1995 A model of the neuro-musculo-skeletal systemfor human locomotion: II. Real-time adaptability undervarious constraints Biol. Cybern. 73 113–21

[42] Taga G 1995 A model of the neuro-musculo-skeletal systemfor human locomotion: I. Emergence of basic gait Biol.Cybern. 73 97–111

[43] Taga G 1998 A model of the neuro-musculo-skeletal systemfor anticipatory adjustment of human locomotion duringobstacle avoidance Biol. Cybern. 78 9–17

[44] Holmes P, Full R J, Koditschek D and Guckenheimer J 2006The dynamics of legged locomotion: models, analyses, andchallenges Siam Rev. 48 207–304

[45] Buchli J, Righetti L and Ijspeert A J 2005 A dynamicalsystems approach to learning: a frequency-adaptive hopperrobot Adv. Artif. Life, Proc. 210–20

[46] Steeves J D and Jordan L M 1980 Localization of a descendingpathway in the spinal cord which is necessary for controlledtreadmill locomotion Neurosci. Lett. 20 283–8

[47] Shik M L, Severin F V and Orlovskii G N 1966 Control ofwalking and running by means of electric stimulation of themidbrain Biofizika 11 659–66

[48] Westerblad H, Lee J A, Lannergren J and Allen D G 1991Cellular mechanisms of fatigue in skeletal muscle Am. J.Physiol. 261 C195–209

[49] Bigland-Ritchie B 1981 EMG/force relations and fatigue ofhuman voluntary contractions Exerc. Sport Sci. Rev.9 75–117

[50] Aerts P, Ker R F, De Clercq D, Ilsley D W and Alexander R M1995 The mechanical properties of the human heel pad: aparadox resolved J. Biomech. 28 1299–308

[51] Ferris D P, Czerniecki J M and Hannaford B 2005 Anankle-foot orthosis powered by artificial pneumatic musclesJ. Appl. Biomech. 21 189–97

[52] Gordon K E, Sawicki G S and Ferris D P 2006Mechanical performance of artificial pneumatic2006muscles to power an ankle-foot orthosis J. Biomech.39 1832–41

[53] Gordon K E and Ferris D P 2007 Learning to walk with arobotic ankle exoskeleton J. Biomech. 40 2636–44

[54] Aaron S L and Stein R B 1976 Comparison of anEMG-controlled prosthesis and the normal human bicepsbrachii muscle Am. J. Phys. Med. 55 1–14

[55] Matsuoka K 1987 Mechanisms of frequency and patterncontrol in the neural rhythm generators Biol. Cybern.56 345–53

[56] Winter D A 1984 Pathologic gait diagnosis withcomputer-averaged electromyographic profiles Arch. Phys.Med. Rehabil. 65 393–8

[57] Williamson M M 2003 Oscillators and crank turning:exploiting natural dynamics with a humanoid robot armPhil. Trans. R. Soc. A 361 2207–23

[58] Williamson M M 1999 Robot arm control exploiting naturaldynamics PhD Thesis Massachusetts Institute ofTechnology, Cambridge

[59] Fukuoka Y, Kimura H and Cohen A H 2003 Adaptive dynamicwalking of a quadruped robot on irregular terrain based onbiological concepts Int. J. Robot. Res. 22 187–202

[60] Cavagna G A, Franzetti P, Heglund N C and Willems P 1988The determinants of the step frequency in running, trottingand hopping in man and other vertebrates J. Physiol. 39981–92

[61] Tongue B H 1996 Principles of Vibration (New York: OxfordUniversity Press)

[62] Lichtwark G A and Wilson A M 2005 In vivo mechanicalproperties of the human Achilles tendon during one-leggedhopping J. Exp. Biol. 208 4715–25

[63] Lichtwark G A and Wilson A M 2006 Interactions between thehuman gastrocnemius muscle and the Achilles tendonduring incline, level and decline locomotion J. Exp. Biol.209 4379–88

10

Bioinsp. Biomim. 3 (2008) 026001 E H Pelc et al

[64] Lichtwark G A, Bougoulias K and Wilson A M 2007 Musclefascicle and series elastic element length changes along thelength of the human gastrocnemius during walking andrunning J. Biomech. 40 157–64

[65] Moritz C T and Farley C T 2004 Passive dynamics change legmechanics for an unexpected surface during human hoppingJ. Appl. Physiol. 97 1313–22

[66] Carrier D R, Heglund N C and Earls K D 1994 Variablegearing during locomotion in the human musculoskeletalsystem Science 265 651–3

[67] Biewener A A 1989 Scaling body support in mammals: limbposture and muscle mechanics Science 245 45–8

[68] Hogan N 1985 The mechanics of multi-joint posture andmovement control Biol. Cybernetics 52 315–31

[69] Nichols T R 2002 The contributions of muscles and reflexes tothe regulation of joint and limb mechanics Clin. Orthop.Relat. Res. 403 S43–50

[70] Ting L H and Macpherson J M 2005 A limited set of musclesynergies for force control during a postural taskJ. Neurophysiol. 93 609–13

[71] Lichtwark G A and Wilson A A 2005 Effects of serieselasticity and activation conditions on muscle power outputand efficiency J. Exp. Biol. 208 2845–53

[72] Lichtwark G A and Wilson A M 2007 Is Achilles tendoncompliance optimised for maximum muscle efficiencyduring locomotion? J. Biomech. 40 1768–75

[73] Hof A L 2003 Muscle mechanics and neuromuscular controlJ. Biomech. 36 1031–8

[74] Loeb G E, Brown I E and Cheng E J 1999 A hierarchicalfoundation for models of sensorimotor control Exp. BrainRes. 126 1–18

[75] Nishikawa K et al 2007 Neuromechanics: an integrativeapproach for understanding motor control Integr. Comp.Biol. 47 16–54

[76] Pearson K G 2004 Generating the walking gait:role of sensory feedback Prog. Brain Res. 143 123–9

[77] Dietz V and Duysens J 2000 Significance of load receptorinput during locomotion: a review Gait Posture 11 102–10

[78] Donelan J M and Pearson K G 2004 Contribution of sensoryfeedback to ongoing ankle extensor activity during thestance phase of walking Can. J. Physiol. Pharmacol.82 589–98

[79] Geyer H, Seyfarth A and Blickhan R 2003 Positive forcefeedback in bouncing gaits? Proc. R. Soc. B 270 2173–83

[80] Grillner S, Georgopoulos A P and Jordan L M 1997 Selectionand initiation of motor behaviour Neurons, Networks andMotor Behavior ed P S G Stein, S Grillner, A Selverstonand D G Stuart (Cambridge: MIT Press) pp 3–19

[81] Alexander R M 2003 Modelling approaches in biomechanicsPhil. Trans. R. Soc. 358 1429–35

[82] McGeer T 1990 Passive bipedal running Proc. R. Soc. B240107–34

[83] McGeer T 1990 Passive dynamic walking Int. J. Robot. Res.9 62–82

[84] Collins S, Ruina A, Tedrake R and Wisse M 2005 Efficientbipedal robots based on passive-dynamic walkers Science307 1082–5

[85] Raibert M H and Sutherland I E 1983 Machines that walk Sci.Am. 248 44–53

[86] Blickhan R, Seyfarth A, Geyer H, Grimmer S, Wagner H andGunther M 2007 Intelligence by mechanics Phil. Trans. A365 199–220

11